It's an attribute of "gens", I would say. Here's a justification after the fact: had he said "et pigerrima" it would tie the phrase to "ad oppugnandarum urbium artes rudis" through the adjective "rudis". "Eadem" (as adverb "likewise", or even "the same people" you could argue) emphasizes a new clause linked to "ad militaria opera".

I know the OLD says that it should be translated as "and also," but I just don't understand how "eadem" came to be used adverbially, especially when it still agrees with a noun in number, gender, and case.

It's not adverbial, actually, though it's best to translate it with an adverb in English. Eadem is nominative and agrees with gens. It literally means "the same", and it's being used as a way of reaffirming the previous subject emphatically. The closest English equivalent to the sense is the adverb "likewise". Idem is commonly used in the nominative this way.

"When they were besieged at Mutina and a tribe unskilled in the methods of attacking cities, and likewise [the same] lazy at military operations, sat sluggish[ly] at the base of the untouched walls..."

I don't really understand the last clause, though. I suppose one of those passive infinitives to be an historical infinitive, with coeptum as subject, but I don't know which, or how to make sense of the other. The word coeptum also seems vague in meaning. This is a tricky sentence, to be sure.

I did say above that you could read it in both those exact ways. How do you know it isn't adverbial here, Imber Ranae? Nonnè suprà dixi id illis duobus modis legi posse? Quomodò, Imber Ranae, scis id hîc adverbium non esse?

"simulari coeptum de pace agi" = (supine with passive infinitive of action is like a future infinitive; supinum cum infinitivo actionis passivo est simile infinitivi futuri) "to be going to initiate [proceedings] about peace to be pretended" = "to be pretending to be going to initiate [proceedings] about peace" = "as if they were about to sue for peace", dicam.

vir litterarum wrote:I know the OLD says that it should be translated as "and also," but I just don't understand how "eadem" came to be used adverbially, especially when it still agrees with a noun in number, gender, and case.

Greek has lots of words like that, e.g. ἑκών agrees with the noun in number, gender, and case, but is normally translated "willingly", which are at heart just predicate adjectives, which Greek at least prefers a lot more than English.

About coeptum, isn't Latin one of the languages that when "begin" is used with a passive infinitive, it can also become passive? I read it here as being "simulari coeptum [est] de pace agi" with a (double?) impersonal passive, to mean something like "there began feigning that there was negotiation for peace", i.e. "they began pretending to negotiate for peace." But I'm not sure if that's likely or even possible.

modus.irrealis wrote About coeptum, isn't Latin one of the languages that when "begin" is used with a passive infinitive, it can also become passive? I read it here as being "simulari coeptum [est] de pace agi" with a (double?) impersonal passive, to mean something like "there began feigning that there was negotiation for peace", i.e. "they began pretending to negotiate for peace." But I'm not sure if that's likely or even possible.

See the note on this page by Zumpt for the usage of the passive forms of "coepi"